LOCAL LARGE DEVIATION THEOREM FOR SUMS OF I.I.D. RANDOM
VECTORS WHEN THE CRAMÉR CONDITION HOLDS IN THE WHOLE
SPACE
Dorota Juszczak
Aleksander V. Nagaev
Abstract: A class of multidimensional distributions is considered. This class contains all the
elliptically contoured distributions having sup-exponential weight function. Each
representative of the class determines a family of the so-called exponential or conjugate
distributions. It is established that the conjugate distribution is asymptotically normal. On the
basis of this normality a large deviation local limit theorem is proved. The theorem assumes
no restrictions on the order of deviations.
2000 AMS Mathematics Subject Classification: Primary: 60F10; Secondary: 26A12,
26B30.
Key words and phrases: Abel theorem, conjugate density, Laplace method, slow
variation.